Hanggao Xin
Shi-Min Hu
Editorial Notes
The authors have requested minor, non-substantive changes to the Version of Record and, in accordance with ACM policies, a Corrected Version of Record (CVoR) was published on August 16, 2022. The authors provided an incorrect funder ID number in the original published version of record. The CVoR corrects this error. For reference purposes, the VoR may still be accessed via the Supplemental Material section on this page.
Abstract
Spherical Harmonics (SH) have been proven as a powerful tool for rendering, especially in real-time applications such as Precomputed Radiance Transfer (PRT). Spherical harmonics are orthonormal basis functions and are efficient in computing dot products. However, computations of triple product and multiple product operations are often the bottlenecks that prevent moderately high-frequency use of spherical harmonics. Specifically state-of-the-art methods for accurate SH triple products of order n have a time complexity of O(n5), which is a heavy burden for most real-time applications. Even worse, a brute-force way to compute k-multiple products would take O(n2k) time. In this paper, we propose a fast and accurate method for spherical harmonics triple products with the time complexity of only O(n3), and further extend it for computing k-multiple products with the time complexity of O(kn3 + k2n2 log(kn)). Our key insight is to conduct the triple and multiple products in the Fourier space, in which the multiplications can be performed much more efficiently. To our knowledge, our method is theoretically the fastest for accurate spherical harmonics triple and multiple products. And in practice, we demonstrate the efficiency of our method in rendering applications including mid-frequency relighting and shadow fields.
Supplemental Material
Available for Download
Version of Record for "Fast and accurate spherical harmonics products" by Xin et al., ACM Transactions on Graphics, Volume 40, Issue 6 (TOG 40:6).
- E Oran Brigham. 1988. The fast Fourier transform and its applications. Prentice-Hall, Inc. Google Scholar
Digital Library
- Taco S. Cohen, Mario Geiger, Jonas Köhler, and Max Welling. 2018. Spherical CNNs. In International Conference on Learning Representations.Google Scholar
- Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, and Kostas Daniilidis. 2018. Learning so(3) equivariant representations with spherical cnns. In Proceedings of the European Conference on Computer Vision. 52--68.Google ScholarCross Ref
- Xuejun Hao and Amitabh Varshney. 2004. Real-time rendering of translucent meshes. ACM Transactions on Graphics 23, 2 (2004), 120--142. Google ScholarDigital Library
- DJ Hofsommer and ML Potters. 1960. Table of Fourier coefficients of associated Legendre functions. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences 63, 5 (1960), 460--480.Google Scholar
- Teturo Inui, Yukito Tanabe, and Yositaka Onodera. 1999. Group theory and its applications in physics. Vol. 78. Springer Verlag.Google Scholar
- Jan Kautz, Peter-Pike Sloan, and Jaakko Lehtinen. 2005. Precomputed Radiance Transfer: Theory and Practice. In ACM SIGGRAPH 2005 Courses. 1--es. Google ScholarDigital Library
- Jan Kautz, John Snyder, and Peter-Pike Sloan. 2002. Fast Arbitrary BRDF Shading for Low-Frequency Lighting Using Spherical Harmonics. Rendering Techniques 2, 291-296 (2002), 1. Google ScholarDigital Library
- Jaakko Lehtinen. 2007. A framework for precomputed and captured light transport. ACM Transactions on Graphics 26, 4 (2007), 13--es. Google ScholarDigital Library
- Jaakko Lehtinen and Jan Kautz. 2003. Matrix radiance transfer. In Proceedings of the 2003 symposium on Interactive 3D graphics. 59--64. Google ScholarDigital Library
- Christian Lessig, Mathieu Desbrun, and Eugene Fiume. 2014. A constructive theory of sampling for image synthesis using reproducing kernel bases. ACM Transactions on Graphics 33, 4 (2014), 1--14. Google ScholarDigital Library
- Xinguo Liu, Peter-Pike J Sloan, Heung-Yeung Shum, and John Snyder. 2004. All-Frequency Precomputed Radiance Transfer for Glossy Objects. Rendering Techniques 2004 (2004). Google ScholarDigital Library
- Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. 2020. NeRF: Representing scenes as neural radiance fields for view synthesis. In European conference on computer vision. 405--421.Google ScholarDigital Library
- Ren Ng, Ravi Ramamoorthi, and Pat Hanrahan. 2003. All-frequency shadows using non-linear wavelet lighting approximation. In ACM SIGGRAPH 2003 Papers. 376--381. Google ScholarDigital Library
- Ren Ng, Ravi Ramamoorthi, and Pat Hanrahan. 2004. Triple product wavelet integrals for all-frequency relighting. In ACM SIGGRAPH 2004 Papers. 477--487. Google ScholarDigital Library
- Derek Nowrouzezahrai, Patricio D. Simari, and Eugene Fiume. 2012. Sparse zonal harmonic factorization for efficient SH rotation. ACM Transactions on Graphics 31, 3 (2012), 23:1--23:9. Google ScholarDigital Library
- Jacopo Pantaleoni, Luca Fascione, Martin Hill, and Timo Aila. 2010. Pantaray: fast ray-traced occlusion caching of massive scenes. ACM Transactions on Graphics 29, 4 (2010), 1--10. Google ScholarDigital Library
- Adrien Poulenard, Marie-Julie Rakotosaona, Yann Ponty, and Maks Ovsjanikov. 2019. Effective rotation-invariant point cnn with spherical harmonics kernels. In 2019 International Conference on 3D Vision. 47--56.Google ScholarCross Ref
- Ravi Ramamoorthi. 2009. Precomputation-based rendering. Foundations and Trends in Computer Graphics and Vision 3, 4 (2009), 281--369. Google ScholarDigital Library
- Ravi Ramamoorthi and Pat Hanrahan. 2001. An Efficient Representation for Irradiance Environment Maps. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '01). 497--500. Google ScholarDigital Library
- Zhong Ren, Rui Wang, John Snyder, Kun Zhou, Xinguo Liu, Bo Sun, Peter-Pike Sloan, Hujun Bao, Qunsheng Peng, and Baining Guo. 2006. Real-time soft shadows in dynamic scenes using spherical harmonic exponentiation. In ACM SIGGRAPH 2006 Papers. 977--986. Google ScholarDigital Library
- Nathanaël Schaeffer. 2013. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochemistry, Geophysics, Geosystems 14, 3 (2013), 751--758.Google ScholarCross Ref
- Peter-Pike Sloan. 2008. Stupid spherical harmonics (sh) tricks. In Game developers conference, Vol. 9. 42.Google Scholar
- Peter-Pike Sloan, Jesse Hall, John Hart, and John Snyder. 2003a. Clustered principal components for precomputed radiance transfer. ACM Transactions on Graphics 22, 3 (2003), 382--391. Google ScholarDigital Library
- Peter-Pike Sloan, Jan Kautz, and John Snyder. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. In Proceedings of the 29th annual conference on Computer graphics and interactive techniques. 527--536. Google ScholarDigital Library
- Peter-Pike Sloan, Xinguo Liu, Heung-Yeung Shum, and John Snyder. 2003b. Bi-scale radiance transfer. ACM Transactions on Graphics 22, 3 (2003), 370--375. Google ScholarDigital Library
- Peter-Pike Sloan, Ben Luna, and John Snyder. 2005. Local, deformable precomputed radiance transfer. ACM Transactions on Graphics 24, 3 (2005), 1216--1224. Google ScholarDigital Library
- Nico Sneeuw and Richard Bun. 1996. Global spherical harmonic computation by two-dimensional Fourier methods. Journal of Geodesy 70, 4 (1996), 224--232.Google ScholarCross Ref
- John Snyder. 2006. Code generation and factoring for fast evaluation of low-order spherical harmonic products and squares. MSR-TR-2006-53 (May 2006), 9.Google Scholar
- Bo Sun and Ravi Ramamoorthi. 2009. Affine double-and triple-product wavelet integrals for rendering. Transactions on Graphics 28, 2 (2009), 1--17. Google ScholarDigital Library
- Weifeng Sun and Amar Mukherjee. 2006. Generalized Wavelet Product Integral for Rendering Dynamic Glossy Objects. In ACM SIGGRAPH 2006 Papers. 955--966. Google ScholarDigital Library
- Justus Thies, Michael Zollhöfer, and Matthias Nießner. 2019. Deferred Neural Rendering: Image Synthesis Using Neural Textures. ACM Transactions on Graphics. 38, 4 (2019), 12 pages. Google ScholarDigital Library
- Karvel K Thornber and David W Jacobs. 2006. Broadened-specular reflection and linear subspaces for object recognition. US Patent 7,058,217.Google Scholar
- Yu-Ting Tsai and Zen-Chung Shih. 2006. All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation. ACM Transactions on Graphics 25, 3 (2006), 967--976. Google ScholarDigital Library
- Jingwen Wang and Ravi Ramamoorthi. 2018. Analytic Spherical Harmonic Coefficients for Polygonal Area Lights. ACM Transactions on Graphics 37, 4 (2018). Google ScholarDigital Library
- Jiaping Wang, Peiran Ren, Minmin Gong, John Snyder, and Baining Guo. 2009. All-frequency rendering of dynamic, spatially-varying reflectance. In ACM SIGGRAPH Asia 2009 papers. 1--10. Google ScholarDigital Library
- Jiaping Wang, Kun Xu, Kun Zhou, Stephen Lin, Shimin Hu, and Baining Guo. 2006. Spherical Harmonics Scaling. The Visual Computer 22 (Sept 2006), 713--720. Google ScholarDigital Library
- Lifan Wu, Guangyan Cai, Shuang Zhao, and Ravi Ramamoorthi. 2020. Analytic spherical harmonic gradients for real-time rendering with many polygonal area lights. ACM Transactions on Graphics 39, 4 (2020), 134--1.Google ScholarDigital Library
- Kun Xu, Yue Gao, Yong Li, Tao Ju, and Shi-Min Hu. 2007. Real-time homogenous translucent material editing. In Computer Graphics Forum, Vol. 26. Wiley Online Library, 545--552.Google Scholar
- Kun Xu, Li-Qian Ma, Bo Ren, Rui Wang, and Shi-Min Hu. 2011. Interactive hair rendering and appearance editing under environment lighting. ACM Transactions on Graphics 30, 6 (2011), 1--10. Google ScholarDigital Library
- Kun Xu, Wei-Lun Sun, Zhao Dong, Dan-Yong Zhao, Run-Dong Wu, and Shi-Min Hu. 2013. Anisotropic Spherical Gaussians. ACM Transactions on Graphics 32, 6 (2013), 209:1--209:11. Google ScholarDigital Library
- Tomoya Yamaguchi, Tatsuya Yatagawa, Yusuke Tokuyoshi, and Shigeo Morishima. 2020. Real-time rendering of layered materials with anisotropic normal distributions. Computational Visual Media 6, 1 (2020), 29--36.Google ScholarCross Ref
- Ling-Qi Yan, Yahan Zhou, Kun Xu, and Rui Wang. 2012. Accurate Translucent Material Rendering under Spherical Gaussian Lights. Computer Graphics Forum 31, 7 (2012), 2267--2276. Google ScholarDigital Library
- Alex Yu, Ruilong Li, Matthew Tancik, Hao Li, Ren Ng, and Angjoo Kanazawa. 2021. Plenoctrees for real-time rendering of neural radiance fields. arXiv preprint arXiv:2103.14024 (2021).Google Scholar
- Kun Zhou, Yaohua Hu, Stephen Lin, Baining Guo, and Heung-Yeung Shum. 2005. Precomputed shadow fields for dynamic scenes. In ACM SIGGRAPH 2005 Papers. 1196--1201. Google ScholarDigital Library
Index Terms
- Fast and accurate spherical harmonics products
Recommendations
Noise-Resistant Fitting for Spherical Harmonics
Spherical harmonic (SH) basis functions have been widely used for representing spherical functions in modeling various illumination properties. They can compactly represent low-frequency spherical functions. However, when the unconstrained least square ...
Efficient multiple scattering in hair using spherical harmonics
Previous research has shown that a global multiple scattering simulation is needed to achieve physically realistic renderings of hair, particularly light-colored hair with low absorption. However, previous methods have either sacrificed accuracy or have ...
Real-time triple product relighting using spherical local-frame parameterization
This paper addresses the problem of real-time rendering for objects with complex materials under varying all-frequency illumination and changing view. Our approach extends the triple product algorithm by using local-frame parameterization, spherical ...
Comments